In Theorem \ref{thm:personalized_lp}, we showed that any personalized
equilibrium is a solution to a linear program plus additional $\min$
constraints, in which all coefficients are rational.  By Theorem
\ref{thm:exist}, this program has at least one solution. Now, we can
rewrite this as a union of many linear programs as follows. Let $F_1,
\ldots, F_{\alpha}$ be the set of all improvement sets. We can write
$\prod_{i=1}^{\alpha} |F_i|$ linear programs, each consisting of the first
three constraints from program \ref{lp_with_mins} as well as the
$\alpha$ constraints [$e_1 = 0$ for some $e_1 \in F_1$], [$e_2 = 0$
  for some $e_2 \in F_2$], $\ldots$, [$e_{\alpha} = 0$ for some
  $e_{\alpha}$ in $F_{\alpha}$]. We can create one LP for each
such combination of one edge from each improvement set, or
$\prod_{i=1}^{\alpha} |F_i|$ LPs. Since the union of these linear programs
is exactly the same as the program in Theorem
\ref{thm:personalized_lp}, and since (by Theorem \ref{thm:exist}) the
program in Theorem \ref{thm:personalized_lp} has at least one
solution, we know that at least one of these linear programs has a
solution. Any feasible LP with rational coefficients will have a
rational solution. Therefore, there will be a personalized equilibrium
with all rational weights.
